A dog is chasing a cat; the cat's velocity is "v".

The dog's velocity is directed towards the cat at all times.
The dog maintains a constant speed "v".
At the instant when the velocity of the dogs "v" and the cats "v" are perpendicular the distance between them is "x".
What's the dogs acceleration, "a", at that moment, and the dog never changes speed

Acceleration is the velocity change per second, or v/s. If both time (undefined) and speed ("never changes") are constant, then there is no change in acceleration. My guess would be 0.

how many electrons does each group lose or gain?

To determine how many electrons each group loses or gains, we need to know which specific elements or groups you are referring to. Different elements have different numbers of electrons in their outer shells, and they gain or lose electrons to achieve a stable electron configuration.

For example, in Group 1 of the periodic table (alkali metals), such as sodium (Na), the outermost electron is easily lost, resulting in a +1 charge. In Group 17 (halogens), such as chlorine (Cl), they gain one electron to achieve a full outer shell, resulting in a -1 charge.

If you could provide more information about the specific groups or elements you are referring to, I can provide a more detailed answer.